Cakewalk Sampling

We study the task of finding good local optima in combinatorial optimization problems. Although combinatorial optimization is NP-hard in general, locally optimal solutions are frequently used in practice. Local search methods however typically converge to a limited set of optima that depend on their initialization. Sampling methods on the other hand can access any valid solution, and thus can be used either directly or alongside methods of the former type as a way for finding good local optima. Since the effectiveness of this strategy depends on the sampling distribution, we derive a robust learning algorithm that adapts sampling distributions towards good local optima of arbitrary objective functions. As a first use case, we empirically study the efficiency in which sampling methods can recover locally maximal cliques in undirected graphs. Not only do we show how our adaptive sampler outperforms related methods, we also show how it can even approach the performance of established clique algorithms. As a second use case, we consider how greedy algorithms can be combined with our adaptive sampler, and we demonstrate how this leads to superior performance in k-medoid clustering. Together, these findings suggest that our adaptive sampler can provide an effective strategy to combinatorial optimization problems that arise in practice.Comment: Accepted as a conference paper by AAAI-2020 (oral presentation

Experimental Evaluation of Approximation and Heuristic Algorithms for Maximum Distance-Bounded Subgraph Problems

In this paper, we consider two distance-based relaxed variants of the maximum clique problem (Max Clique), named Maxd-Clique and Maxd-Club for positive integers d. Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of n1−ε for any real ε>0 unless P=NP , since they are identical to Max Clique (Håstad in Acta Math 182(1):105–142, 1999; Zuckerman in Theory Comput 3:103–128, 2007). In addition, it is NP -hard to approximate Maxd-Clique and Maxd-Club to within a factor of n1/2−ε for any fixed integer d≥2 and any real ε>0 (Asahiro et al. in Approximating maximum diameter-bounded subgraphs. In: Proc of LATIN 2010, Springer, pp 615–626, 2010; Asahiro et al. in Optimal approximation algorithms for maximum distance-bounded subgraph problems. In: Proc of COCOA, Springer, pp 586–600, 2015). As for approximability of Maxd-Clique and Maxd-Club, a polynomial-time algorithm, called ReFindStar d, that achieves an optimal approximation ratio of O(n1/2) for Maxd-Clique and Maxd-Club was designed for any integer d≥2 in Asahiro et al. (2015, Algorithmica 80(6):1834–1856, 2018). Moreover, a simpler algorithm, called ByFindStar d, was proposed and it was shown in Asahiro et al. (2010, 2018) that although the approximation ratio of ByFindStar d is much worse for any odd d≥3, its time complexity is better than ReFindStar d. In this paper, we implement those approximation algorithms and evaluate their quality empirically for random graphs. The experimental results show that (1) ReFindStar d can find larger d-clubs (d-cliques) than ByFindStar d for odd d, (2) the size of d-clubs (d-cliques) output by ByFindStar d is the same as ones by ReFindStar d for even d, and (3) ByFindStar d can find the same size of d-clubs (d-cliques) much faster than ReFindStar d. Furthermore, we propose and implement two new heuristics, Hclub d for Maxd-Club and Hclique d for Maxd-Clique. Then, we present the experimental evaluation of the solution size of ReFindStar d, Hclub d, Hclique d and previously known heuristic algorithms for random graphs and Erdős collaboration graphs

Brokerage discovery in social networks

Conferência realizada em Angra do Heroísmo, Açores, de 9-12 de Setembro de 2013In social networks two types of measures can be identified, the structural measures and community structure based on diameter and centrality. The community structure usually deals with network partition into communities. The key idea of this work is to explore the concept of strong and weak ties by finding brokers within communities. The strict partition problem is relaxed into a bi-objective set covering problem with k-cliques which allows over-covered and uncovered nodes. The information extracted from social networking goes beyond cohesive groups, allowing the finding of brokers that interact between groups

Low-Diameter Clusters in Network Analysis

In this dissertation, we introduce several novel tools for cluster-based analysis of complex systems and design solution approaches to solve the corresponding optimization problems. Cluster-based analysis is a subfield of network analysis which utilizes a graph representation of a system to yield meaningful insight into the system structure and functions. Clusters with low diameter are commonly used to characterize cohesive groups in applications for which easy reachability between group members is of high importance. Low-diameter clusters can be mathematically formalized using a clique and an s-club (with relatively small values of s), two concepts from graph theory. A clique is a subset of vertices adjacent to each other and an s-club is a subset of vertices inducing a subgraph with a diameter of at most s. A clique is actually a special case of an s-club with s = 1, hence, having the shortest possible diameter. Two topics of this dissertation focus on graphs prone to uncertainty and disruptions, and introduce several extensions of low-diameter models. First, we introduce a robust clique model in graphs where edges may fail with a certain probability and robustness is enforced using appropriate risk measures. With regard to its ability to capture underlying system uncertainties, finding the largest robust clique is a better alternative to the problem of finding the largest clique. Moreover, it is also a hard combinatorial optimization problem, requiring some effective solution techniques. To this aim, we design several heuristic approaches for detection of large robust cliques and compare their performance. Next, we consider graphs for which uncertainty is not explicitly defined, studying connectivity properties of 2-clubs. We notice that a 2-club can be very vulnerable to disruptions, so we enhance it by reinforcing additional requirements on connectivity and introduce a biconnected 2-club concept. Additionally, we look at the weak 2-club counterpart which we call a fragile 2-club (defined as a 2-club that is not biconnected). The size of the largest biconnected 2-club in a graph can help measure overall system reachability and connectivity, whereas the largest fragile 2-club can identify vulnerable parts of the graph. We show that the problem of finding the largest fragile 2-club is polynomially solvable whereas the problem of finding the largest biconnected 2-club is NP-hard. Furthermore, for the former, we design a polynomial time algorithm and for the latter - combinatorial branch-and-bound and branch-and-cut algorithms. Lastly, we once again consider the s-club concept but shift our focus from finding the largest s-club in a graph to the problem of partitioning the graph into the smallest number of non-overlapping s-clubs. This problem cannot only be applied to derive communities in the graph, but also to reduce the size of the graph and derive its hierarchical structure. The problem of finding the minimum s-club partitioning is a hard combinatorial optimization problem with proven complexity results and is also very hard to solve in practice. We design a branch-and-bound combinatorial optimization algorithm and test it on the problem of minimum 2-club partitioning